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BetterGrades Precalculus · Unit 15 · Unit map

Sequences, Series, and Discrete Models

The first five terms are 4,7,12,19,28. Plot them as a discrete function and propose a formula.

Precalculus · Unit 15

The lesson path

The first five terms are 4,7,12,19,28. Plot them as a discrete function and propose a formula.

Core sequence

Sequences, Series, and Discrete Models

Move in order when the material is new, or enter at the exact skill you need.

  1. 01Sequences as discrete functionsInterpret a sequence as a function on an integer index set.
  2. 02Explicit and recursive descriptionsMove among term lists, explicit formulas, recursive rules, tables, and plots.
  3. 03Arithmetic sequencesDerive and use an=a1+(n−1)da_n=a_1+(n-1)d and connect arithmetic sequences to sampled linear functions.
  4. 04Geometric sequencesDerive and use an=a1rn−1a_n=a_1r^{n-1} and connect geometric sequences to sampled exponential functions.
  5. 05Recurrence, iteration, and discrete dynamical modelsAnalyze repeated function application, fixed points, cycles, and qualitative long-run behavior.
  6. 06Sigma notation and finite sumsInterpret summation notation, index bounds, summands, and term counts.
  7. 07Arithmetic seriesDerive and use Sn=n(a1+an)2S_n=\frac{n(a_1+a_n)}{2}.
  8. 08Finite geometric seriesDerive and use Sn=a1(1−rn)1−rS_n=\frac{a_1(1-r^n)}{1-r}.
  9. 09Infinite geometric series and convergenceDetermine convergence and evaluate infinite geometric sums when ∣r∣<1|r|<1.
  10. 10Mathematical inductionProve statements for all integers in a domain using base case and inductive step.
  11. 11Pascal's triangle and binomial coefficientsConnect combinations, recursive construction, symmetry, and polynomial coefficients.
  12. 12The binomial theorem and discrete-model synthesisExpand binomial powers and compare arithmetic, geometric, recursive, and accumulated models.
Source record

References used for this unit

  • Stitz & Zeager, Precalculus, Chapter 9
  • University of Washington Precalculus, discrete-model problems
  • AP Precalculus framework, sequence and model connections

The learner copy is original BetterGrades material informed by these rights-separated references; no long source passage is reproduced.